Definition:Semiring (Abstract Algebra)
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This page is about Semiring in the context of Abstract Algebra. For other uses, see Semiring.
Definition
A semiring is a ringoid $\struct {S, *, \circ}$ in which:
That is, such that $\struct {S, *, \circ}$ has the following properties:
| \((\text A 0)\) | $:$ | \(\ds \forall a, b \in S:\) | \(\ds a * b \in S \) | ||||||
| \((\text A 1)\) | $:$ | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a * b} * c = a * \paren {b * c} \) | ||||||
| \((\text M 0)\) | $:$ | \(\ds \forall a, b \in S:\) | \(\ds a \circ b \in S \) | ||||||
| \((\text M 1)\) | $:$ | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | ||||||
| \((\text D)\) | $:$ | \(\ds \forall a, b, c \in S:\) | \(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \) | ||||||
| \(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \) |
These are called the semiring axioms.
Also defined as
There are various other conventions on what constitutes a semiring.
Some of these have a distinguished, different name on $\mathsf{Pr} \infty \mathsf{fWiki}$:
- An additive semiring is a semiring whose distributand is commutative
- A rig is a semiring whose distributand forms a commutative monoid
Still, some sources impose further that there be a identity element for the distributor, that is, that $\struct {S, \circ}$ be a monoid.
Such a structure could be referred to as a rig with unity, consistent with the definition of ring with unity.
This website thus specifically defines a semiring as one fulfilling axioms $\text A 0, \text A 1, \text M 0, \text M 1, \text D$ only (that is, as two semigroups bound by distributivity).
Also see
Examples
Stronger properties
- Definition:Commutative Ring with Unity
- Definition:Commutative Ring
- Definition:Ring (Abstract Algebra)
- Definition:Commutative Semiring
- Definition:Rig
- Definition:Additive Semiring